Problem 28 · 2011 Math Kangaroo
Stretch
Geometry & Measurement
area-decomposition
The two circles shown intersect each other at X and Y. Here XY is the diameter of the small circle. The centre S of the large circle (with radius r) lies on the small circle. How big is the area of the grey region?
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Answer: C — \(\dfrac{1}{2}r^2\)
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Hint 1 of 2
Since S lies on the small circle and XY is its diameter, angle XSY is a right angle.
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Hint 2 of 2
The grey lune's area equals the area of the right triangle XSY.
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Approach: use the lune = triangle identity
- XY is the small circle's diameter and S lies on that circle, so angle XSY = 90°.
- X and Y are on the large circle, so SX = SY = r, making XSY a right isosceles triangle of area ½r².
- By the classic lune result, the grey crescent has the same area as triangle XSY.
- So the grey area is ½r², choice (C).
Mark:
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